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A circle is centered at the point [tex]$(5,-4)$[/tex] and passes through the point [tex]$(-3,2)$[/tex].

The equation of this circle is [tex]\((x + \square)^2 + (y + \square)^2 = \square\)[/tex].


Sagot :

To determine the equation of a circle with a given center and a point on the circle, we need to identify three parameters in the standard form of the circle's equation:

[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]

where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.

Step-by-step solution:

1. Identify the center of the circle:
The center of the circle is given as [tex]\((h, k) = (5, -4)\)[/tex].

2. Identify a point on the circle:
The point on the circle is [tex]\((x_1, y_1) = (-3, 2)\)[/tex].

3. Calculate the radius:
The radius [tex]\(r\)[/tex] can be found using the distance formula between the center [tex]\((5, -4)\)[/tex] and the given point [tex]\((-3, 2)\)[/tex]:

[tex]\[ r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} \][/tex]

Plugging in the values:

[tex]\[ r = \sqrt{((-3 - 5)^2 + (2 + 4)^2)} = \sqrt{((-8)^2 + (6)^2)} = \sqrt{64 + 36} = \sqrt{100} = 10 \][/tex]

4. Form the equation of the circle:
Using the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], and substituting [tex]\(h = 5\)[/tex], [tex]\(k = -4\)[/tex], and [tex]\(r = 10\)[/tex], the equation becomes:

[tex]\[ (x - 5)^2 + (y + 4)^2 = 10^2 = 100 \][/tex]

5. Fit the equation into the required format:
The standard form given in the question is [tex]\((x + \square)^2 + (y + \square)^2 = \square\)[/tex].

To match this, consider:

[tex]\[ (x - 5)^2 = (x + (-5))^2 \][/tex]
and
[tex]\[ (y + 4)^2 = (y + (-4))^2. \][/tex]

6. Final values for the equation:
Substitute the appropriate values into the given format:

[tex]\[ (x + (-5))^2 + (y + (-4))^2 = 100 \][/tex]

Thus, the correct values to fill in the blanks are:
[tex]\((x + \boxed{-5})^2 + (y + \boxed{-4})^2 = \boxed{100}\)[/tex]